Schubert Objects

Abstract

Schubert polynomials arose from questions involving enumerative and algebraic geometry, representation theory, and algebraic topology. They have been studied from a variety of perspectives, each with its own combinatorial object [1, 4, 5, 17, 15, 16, 28, 50, 66]. In this dissertation, the combinatorial objects which index the monomials in a Schubert polynomial are called Schubert objects. There are many such objects and one of the main goals of this dissertation is to illumination the bijections between them. In addition to exploring the bijections between Schubert objects, we explore different methods of constructing them. The construction methods are all developed using trees of Schubert objects and taking the collection of leaves at the end of the tree. We introduce a new method to compute the decomposition of Schubert polynomials into key polynomials. We also define a new operator, called split, which provides an alternative approach of creating a tree of rc-graphs. A new Schubert object is explored, called an inversion filling. We discuss a special case of inversion fillings, the Grassmannian permutation case, which gives rise to a left divided difference operator on semistandard Young tableaux. In addition, we describe the previously known construction of skyline fillings and their connection to other Schubert objects.